TIDALLY DRIVEN ROCHE-LOBE OVERFLOW OF HOT
JUPITERS WITH MESA
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Valsecchi, Francesca, Saul Rappaport, Frederic A. Rasio, Pablo
Marchant, and Leslie A. Rogers. “TIDALLY DRIVEN ROCHELOBE OVERFLOW OF HOT JUPITERS WITH MESA.” The
Astrophysical Journal 813, no. 2 (November 3, 2015): 101. ©
2015 The American Astronomical Society
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http://dx.doi.org/10.1088/0004-637x/813/2/101
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The Astrophysical Journal, 813:101 (13pp), 2015 November 10
doi:10.1088/0004-637X/813/2/101
© 2015. The American Astronomical Society. All rights reserved.
TIDALLY DRIVEN ROCHE-LOBE OVERFLOW OF HOT JUPITERS WITH MESA
Francesca Valsecchi1, Saul Rappaport2, Frederic A. Rasio1, Pablo Marchant3, and Leslie A. Rogers4
1
Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), and Northwestern University, Department of Physics and Astronomy,
Evanston, IL 60208, USA; francesca@u.northwestern.edu, rasio@northwestern.edu
2
Department of Physics, and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA; sar@mit.edu
3
Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hgel 71, D-53121 Bonn, Germany; pablo@astro.uni-bonn.de
4
Department of Astronomy and Department of Geophysics and Planetary Sciences, California Institute of Technology,
Pasadena, CA 91125, USA; larogers@caltech.edu
Received 2015 June 16; accepted 2015 September 29; published 2015 November 3
ABSTRACT
Many exoplanets have now been detected in orbits with ultra-short periods very close to the Roche limit. Building
upon our previous work, we study the possibility that mass loss through Roche lobe overflow (RLO) may affect the
evolution of these planets, and could possibly transform a hot Jupiter into a lower-mass planet (hot Neptune or
super-Earth). We focus here on systems in which the mass loss occurs slowly (“stable mass transfer” in the
language of binary star evolution) and we compute their evolution in detail with the binary evolution code Modules
for Experiments in Stellar Astrophysics. We include the effects of tides, RLO, irradiation, and photo-evaporation
(PE) of the planet, as well as the stellar wind and magnetic braking. Our calculations all start with a hot Jupiter
close to its Roche limit, in orbit around a Sun-like star. The initial orbital decay and onset of RLO are driven by
tidal dissipation in the star. We confirm that such a system can indeed evolve to produce lower-mass planets in
orbits of a few days. The RLO phase eventually ends and, depending on the details of the mass transfer and on the
planetary core mass, the orbital period can remain around a few days for several Gyr. The remnant planets have
rocky cores and some amount of envelope material, which is slowly removed via PE at a nearly constant orbital
period; these have properties resembling many of the observed super-Earths and sub-Neptunes. For these remnant
planets, we also predict an anti-correlation between mass and orbital period; very low-mass planets (Mpl  5 M⊕)
in ultra-short periods (Porb < 1 day) cannot be produced through this type of evolution.
Key words: planet–star interactions – planetary systems – planets and satellites: gaseous planets – stars: evolution –
stars: general
1. INTRODUCTION
However, we note that the host stars of the tightest hot Jupiters
are observed to be rotating slowly at present.
In Valsecchi et al. (2014), we investigated the fate of a hot
Jupiter transferring mass to its stellar host using a simplified
binary MT model. We showed that the planet could be stripped
of its envelope, resulting in a hot super-Earth-type planet. This
model naturally solves some of Keplerʼs evolutionary puzzles
(e.g., Kepler-78; Howard et al. 2013; Pepe et al. 2013; SanchisOjeda et al. 2013), and it could explain the excess of isolated
hot super-Earth- or sub-Neptune-size planets seen in the Kepler
data (Steffen & Farr 2013).
Our previous work relied on several simplifying, and
potentially key, assumptions. First, while using detailed models
for the host star, we adopted published mass–radius relations
for the planetary component, thus assuming that the planet
remains in thermal equilibrium throughout the MT. Second,
even though our planetary models included the effect of stellar
irradiation on the planetary mass–radius relations, irradiation
was kept fixed, while it is expected to vary as the orbit evolves
during MT. Furthermore, we neglected the resulting mass loss
due to photo-evaporation (PE; Murray-Clay et al. 2009;
Jackson et al. 2010; Lopez et al. 2012; Lopez & Fortney 2013;
Owen & Alvarez 2015), even though various studies found it to
play an important role in the evolution of highly irradiated
super-Earth and sub-Neptune-type planets (e.g., Jackson
et al. 2010; Rogers et al. 2011; Lopez et al. 2012; Batygin &
Stevenson 2013; Lopez & Fortney 2013; Owen & Wu 2013).
As a natural continuation of our previous work, here we
significantly expand upon our simple binary MT model to
include detailed planetary evolution, as well as the effects of a
Hot Jupiters, giant planets in orbits of a few days, constitute
one of the many surprises of exoplanet searches. Whether their
tight orbits are the result of inward migration in a protoplanetary disk (Goldreich & Tremaine 1980; Lin et al. 1996;
Ward 1997; Murray et al. 1998) or tidal circularization of an
orbit made highly eccentric via gravitational interactions (Rasio
& Ford 1996; Wu & Murray 2003; Fabrycky & Tremaine 2007;
Chatterjee et al. 2008; Nagasawa et al. 2008; Naoz et al. 2011;
Wu & Lithwick 2011; Plavchan & Bilinski 2013; Valsecchi &
Rasio 2014a, 2014b) is still matter of debate. Certainly,
independent of the formation mechanism, tidal dissipation in
slowly spinning host stars is causing the orbits of the tightest
hot Jupiters currently known to shrink rapidly (e.g., Rasio
et al. 1996; Sasselov 2003; Birkby et al. 2014; Valsecchi &
Rasio 2014a; see also Table 1 in Valsecchi & Rasio 2014a and
references therein).
Eventually, hot Jupiters may decay down to their Rochelimit separation. While it is commonly assumed that the planet
is then quickly accreted by the star (e.g., Jackson et al. 2009;
Metzger et al. 2012; Schlaufman & Winn 2013; Damiani &
Lanza 2014; Teitler & Königl 2014; Zhang & Penev 2014), for
a typical system hosting a hot Jupiter that orbits a Sun-like star,
the ensuing mass transfer (hereafter “MT”) may be dynamically
stable (Sepinsky et al. 2010). This was suggested, e.g., to
explain WASP-12ʼs transit features (Lai et al. 2010). Trilling
et al. (1998) investigated the possibility of halting inward disk
migration through tides and Roche-lobe overflow (“RLO”) MT
from a hot Jupiter to a rapidly spinning (young) stellar host.
1
The Astrophysical Journal, 813:101 (13pp), 2015 November 10
Valsecchi et al.
However, we note that Lopez & Fortney (2014) found such
heating to play an important role in delaying cooling and
contraction, particularly for planets less than 5 M⊕. This could
lead to an underestimate of the radii of sub-Neptune planets,
especially at ages „1 Gyr. Below we focus on planetary
models at least 2 Gyr old.
We account for the effects of irradiation and PE as follows.
For irradiation, we use the F*–Σpl surface heating function.
Here F* is the day-side flux from the star at the substellar point,
given by
Table 1
Definition of the Main Parameters used in This Work
Parameter
Definition
M*, Mpl
Mc, Menv
R*, Rpl
Rlobe,*, Rlobe, pl
T*, Tpl
Ω*, Ωpl (Ωo)
Z
a
Porb
Jorb
f = Menv/Mpl
q = Mpl/M*
t, tMS
Mass
Planetary core and envelope mass
Radius
Roche-lobe radius
Surface temperature
Spin (orbital) frequency
Metallicity
Semimajor axis
Orbital period
Orbital angular momentum
Envelope mass fraction
Planet to star mass ratio
Stellar age and main-sequence lifetime
⎛ R ⎞2
F* = s T 4 ⎜ * ⎟ .
*⎝ a ⎠
The planet equilibrium temperature, Teq, is
⎛ R ⎞1 2
Teq = T* ⎜ * ⎟ ,
⎝ 2a ⎠
Note. The subscripts “*” and “pl” refer to the star and planet, respectively. We
take the stellar age to be equal to the system age, and the main-sequence
lifetime for solar mass stars with radiative cores to be the age when the mass
fraction of H at the center of the star 0.
(2 )
(Saumon et al. 1996) so that the power received from the host
star could be radiated away in equilibrium if the planet had this
temperature over its entire surface. The parameter Σpl is the
column depth reached by irradiation. Here we adopt a value of
Σpl = 1 g cm−2, which yields planetary mass–radius relations
in agreement with detailed models by Fortney et al. (2007),
within a few percent for highly irradiated planets, as shown in
Figure 1. We discuss the effect of varying Σpl in Section 4.1.3.
In what follows, the planetary radius corresponds to an optical
depth τ = 2/3.
Irradiation leads to “PE” mass loss from the planet.
Specifically, this process is thought to be most efficient when
a hot Jupiter is strongly irradiated by ultraviolet (UV) and
X-ray photons, which photo-ionize atomic H in the planetary
atmosphere. The resulting heat input, when high enough to
induce temperatures corresponding to the escape velocity, can
cause outflows. Murray-Clay et al. (2009) identified two
regimes, based on the stellar flux FXUV (however, see Owen &
Alvarez 2015). For large FXUV, like those typical of T-Tauri
stars (105 erg cm−2 s−1) the mass loss is “radiation/recombination” limited and it is described by
varying irradiation and the consequent PE mass loss from the
planet. For these new calculations we utilize the Modules for
Experiments in Stellar Astrophysics (MESA) evolution code
(Paxton et al. 2011, 2013, 2015). The MESA inlist files used in
this work can be downloaded from Valsecchi et al. (2015).
The plan of the paper is as follows. In Section 2 we describe
the stellar and planetary models used in this work. In Section 3
we describe our orbital evolution model and the physical
mechanisms entering the calculation. We present some
examples of our orbital evolution calculations in Section 4
and discuss our results in Section 5. We conclude in Section 6.
For quick reference, the notations adopted in this work are
summarized in Table 1.
2. STELLAR AND PLANETARY MODELS
The stellar and planetary models adopted in this work are
computed
with
MESA
(version
7184;
Paxton
et al. 2011, 2013, 2015). In particular, the planets are created
and evolved closely following the test suite make_planets
provided within MESA and the input files yielding Figure 3 of
Paxton et al. (2013).5 In what follows, we give specifications
only for those MESA parameters that are changed from the
values adopted in these input files.
In all of our calculations, we consider a 1 Me star paired with
a 1 MJ planet. We assume solar composition (Y = 0.27,
Z = 0.02) for both the star and planet envelope, and keep the
mixing length parameter αMLT to MESAʼs default value of 2.
For the planet, we expand on our previous work (Valsecchi
et al. 2014) and consider models with solid cores of masses
Mc = 1 M⊕, 5 M⊕, 10 M⊕, 15 M⊕, and 30 M⊕. For the cores, we
use a constant density of 5 g cm−3, following Batygin &
Stevenson (2013). Furthermore, we take the heat flux arising
from radioactive decay in the core to be zero, following
Fortney et al. (2007). As these authors point out, this is a
common assumption in evolutionary models of Jupiter- and
Saturn-type planets (Hubbard 1977; Saumon et al. 1992;
Fortney & Hubbard 2003), as this approach introduces a small
error compared to other unknowns entering the problem.
5
(1 )
⎞1
⎛
FXUV
M˙ rr ‐ lim ~ 4 ´ 1012 ⎜
⎟
⎝ 5 ´ 10 5 erg cm-2 s-1 ⎠
2
g s-1.
(3 )
For lower FXUV values the mass loss is “energy limited” and it
is described by (Erkaev et al. 2007; Lopez et al. 2012)
3
pFXUV RXUV
M˙ e‐ lim »
,
GMp Ktide
(4 )
where Ktide = 1 − (3/2)(1/ξ) + (1/2)(1/ξ3) and ξ = RHill/
RXUV. Since in all of our calculations FXUV (computed as
described below) remains below 105 erg cm−2 s−1, we use only
Equation (4). However, we discuss whether our results are
sensitive to the PE prescription in Section 4.1.4 and Table 2.
For the flux, we follow Ribas et al. (2005) and take
⎛ t ⎞-1.23 ⎛ a ⎞-2
⎜
⎟
FXUV = 29.7 ⎜
erg s-1 cm-2,
⎟
⎝ AU ⎠
⎝ Gyr ⎠
(5)
where we have scaled their result at 1 AU to an arbitrary
distance. The parameter ò represents the efficiency of
converting FXUV into usable work, while RXUV is the radius
Available at http://mesastar.org/results/mesa2/planets
2
The Astrophysical Journal, 813:101 (13pp), 2015 November 10
Valsecchi et al.
(Lopez & Fortney 2013, hereafter LF13). Here we closely
follow the recent results of LF13 and adopt ò = 0.1. We note,
however, that hydrodynamic calculations suggest that ò can
vary between 0.01 and 0.2, depending on the mass of the planet
(Owen & Jackson 2012). With ò = 0.1, the RXUV value that
better matches the results of detailed numerical calculations by
LF13 for systems 1–10 Gyr old is RXUV = 1.2 Rpl (see below)
and we use this estimate of RXUV throughout our calculations.6
Finally, Ktide is to account for the fact that the mass leaving the
planet needs only to reach the Hill radius to escape, where
RHill = Mpl1 3 (3M* )-1 3a (Erkaev et al. 2007).
We test the implementation of PE by comparing with the
mass-loss calculations presented by LF13 for the planet Kepler36c. These are shown in Figure 2. Following their work, we
create a 9.4 M⊕, 10 Myr old irradiated planet with a core of
7.4 M⊕ and Z = 0.35. We adopt Teq = 930 K (Carter et al. 2012
reports Teq = 928 ± 10 K) and evolve the planetary model with
irradiation (fixed) and PE, according to Equation (4). The
parameters adopted in this work are those yielding agreement
with LF13 within a few percent for 1–10 Gyr old systems
(black solid line).
3. ORBITAL EVOLUTION MODEL
MESA allows us to track the evolution of both the planet and
star simultaneously, while orbital evolution is followed by
taking into account changes to the orbital angular momentum
of the system. Below we describe the physical mechanisms
included in our model: “PE,” tides, magnetic braking (“MB”;
Skumanich 1972), and RLO. For simplicity, we neglect the
effects of stellar wind mass loss. From the wind prescription
provided in MESAʼs test suite 1M_pre_ms_to_wd for the
evolution of a 1 Me star (Reimers 1975, Bloecker 1995), we
find that the orbital evolution timescales associated with stellar
winds are longer than 1012 year throughout the main-sequence
evolution of the host star.
3.1. Photo-evaporation
Mass escape via PE affects the orbital separation and the spin
of the planet. For simplicity, we assume spherically symmetric
mass loss, which carries away the specific angular momentum
of the mass-losing component. With this assumption, PE
affects the orbital angular momentum Jorb according to
⎛ J˙orb ⎞
M˙ pl,PE 1
M˙ pl,PE
,

⎜
⎟ =
⎝ Jorb ⎠PE
Mpl 1 + q
Mpl
Figure 1. Planetary mass–radius relations at 4.5 Gyr for different core masses
and equilibrium temperatures. From top to bottom: coreless planets,
Mc = 10 M⊕ and 25 M⊕, as illustrative examples. The solid lines are the
models by Fortney et al. (2007; from Table 4 of their paper) at an equilibrium
temperature of 78 K (light gray), 1300 K (dark gray), and 1960 K (black). The
data points are the models computed with MESA for Σpl set to 1 g cm−2 (“×”)
and 100 g cm−2 (“,”). Varying Σpl between 0.1 and 1 g cm−2 makes no
significant difference. The value of Σpl is not important for Teq „ 1300 K
(symbols overlap). As in Fortney et al. (2007), the radii correspond to a
pressure of ∼1 bar. We note that the initial value of the planetary radius
required by the create _initial _model routine (see Section 2.1 of Paxton
et al. 2013) was varied between 2 and 5 RJ to facilitate MESAʼs convergence.
(6 )
where q = Mpl/M*. For reference, the evolution of the orbital
separation due to PE is given by
M˙ pl,PE q
⎛ a˙ ⎞
⎜ ⎟
= 0,
⎝ a ⎠PE
Mpl 1 + q
(7 )
PE is always active and M˙ pl,PE is given by either Equation (3)
or (4); depending on FXUV though, in practice, only
Equation (4) is needed in our calculations. As far as the
planetary spin is concerned, PE carries away the angular
momentum of the corresponding shells of material.
of the planet at which the atmosphere becomes optically thick
to XUV photons. Murray-Clay et al. (2009) place RXUV at a
surface pressure of ∼10−9 bar, which corresponds to a radius
that is typically 10%–20% greater than the optical photosphere
6
MESA uses automatic mesh refinement, thus adjusting the number of mesh
points of the planetary model at the beginning of each time step, if necessary.
For this reason, the point at the surface where the pressure is ∼10−9 bar is not
always resolved.
3
The Astrophysical Journal, 813:101 (13pp), 2015 November 10
Valsecchi et al.
Table 2
Summary of Results
Mc
γ
PE
(g cm−2)
(M⊕)
5
10
15
30
5
10
15
30
5
30
5
30
5
30
5
30
Σpl
L
L
L
L
0.6
0.7
0.7
0.7
L
L
0.6
0.7
L
L
0.6
0.7
1
1
1
1
1
1
1
1
1
1
1
1
100
100
100
100
e-lim
e-lim
e-lim
e-lim
e-lim
e-lim
e-lim
e-lim
e-lim + rr-lim
e-lim + rr-lim
e-lim + rr-lim
e-lim + rr-lim
e-lim
e-lim
e-lim
e-lim
ΔtRLO
(Gyr)
Porb at the
End of RLO
(day)
f at the
End of RLO
(%)
Δt When
f ∼ 20% to 30%
(Gyr)
Porb When
f ∼ 20%
(day)
Δt When
f ∼ 7% to 20%
(Gyr)
Porb When
f ∼ 7%
(day)
t When
f ∼ 7%
(tMS)
4.5
4.4
4.2
3.7
1.6
1.2
1.3
2.0
6.9
5.6
3.7
3.0
4.7
4.2
1.7
2.0
3.4
1.6
1.2
0.5
3.9
1.7
1.2
0.5
3.8
0.5
3.5
0.5
3.7
0.5
4.3
0.7
39.8
43.1
43.1
7.0
41.1
46.0
48.2
7.0
37.7
7.0
40.1
7.0
40.2
7.0
41.7
19.4
0.2
0.5
0.6
0.3
0.1
0.3
0.5
0.3
0.2
0.4
0.2
0.4
0.2
0.4
0.2
0.3
3.4
1.6
1.1
0.6
3.9
1.7
1.2
0.6
3.8
0.6
3.5
0.6
3.7
0.7
4.3
0.7
3.1
2.9
2.0
0.3
5.2
3.4
3.4
0.4
1.6
0.3
3.8
0.3
3.5
0.4
5.3
0.6
3.4
1.6
0.9
0.5
3.9
1.7
1.1
0.5
3.8
0.5
3.5
0.5
3.7
0.5
4.3
0.5
1.0
1.0
0.9
0.6
0.9
0.7
0.8
0.4
1.1
0.8
1.0
0.5
1.0
0.6
1.0
0.5
Note. The results include f = Menv/Mpl values down to 7% (but for the values in boldface where f … 8%), to allow a comparison between different MT assumptions
and Mc values. In fact, for some of the models considered MESA does not converge when Menv/Mpl drops below about 7%. Time intervals are denoted with Δt.
Column PE denotes the photo-evaporation prescription, with e-lim and rr-lim indicating the energy-limited and radiation/recombination-limited regimes, respectively.
The parameter t denote the age of the system in units of the stellar main-sequence lifetime. The first four examples are for conservative MT. For non-conservative MT
we use δ = 1 and vary γ as summarized in the second column. Note, the 30 M⊕ core cases never detaches, but for the last case (non-conservative MT evolution with
Σpl 100 g cm−2). In this case, 0.5 Gyr after the first 2 Gyr long RLO phase, the planet fills its Roche lobe again.
3.2. Tides
Tides affect the stellar and planetary spins, as well as the
orbital separation, transferring angular momentum between the
orbit and the components’ spin. Within MESA, tides are first
applied to the spins. The orbital separation is then varied so as
to conserve total angular momentum. Here we take the
components to be rotating as solid bodies (however, see
Stevenson 1979; Barker et al. 2014).
For stellar tides we proceed as in Valsecchi et al. (2014) and
Valsecchi & Rasio (2014a, 2014b). Specifically, we adopt the
weak-friction approximation (Zahn 1977, 1989) using a
parametrization for tidal dissipation calibrated from observations of stellar binaries, as in Hurley et al. (2002). This assumes
that tides are dissipated in the stellar convection zone via eddy
viscosity. Furthermore, we reduce the efficiency of tides at high
tidal forcing frequencies (when the forcing frequency is higher
than the convective turnover frequency of the largest eddies)
linearly, following Zahn (1966), and as suggested by recent
numerical results by Penev et al. (2007).
For planetary tides, we assume that they efficiently maintain
the planet in a tidally locked configuration (Ωpl = Ωo)
throughout the evolution. In fact, for a tidal quality factor
Q′ = 106 (typical for gas giants) and a nearly synchronized
planet,7 the spin synchronization timescale due to static tides in
the planet is ∼2–4 orders of magnitude shorter than the
timescale related to the main driver of the orbital evolution (i.e.,
mass loss from the planet; see Section 4), depending on the
core mass. Clearly, tides would synchronize the planet even
faster for lower values of Q′ more appropriate for rocky
Figure 2. Mass-loss evolution of Kepler-36c. Percent of mass contained in the
envelope as a function of time for different RXUV and ò values. The gray data
points are taken from Figure 1 of LF13. The parameter Σpl was set to 1 g cm−2.
For FXUV, we follow Ribas et al. (2005), but rescale the flux at the Kepler-36c
orbital separation (a = 0.12 AU). As in LF13, we keep FXUV constant at the
100 Myr value when the star is younger than 100 Myr and we let it evolve
when the star is older than 100 Myr. Here we assume a 1 Me companion. At
the surface (τ = 2/3), the pressure is ∼10 mbar (where LF13 place the
transiting radius).
While we assume that mass loss is spherically symmetric, we
note that strong magnetic fields may confine the flow primarily
to the poles and day-side of the planet (Owen & Adams 2014).
Teyssandier et al. (2015) studied the torque on super-Earth and
sub-Neptune-type planets due to anisotropic photo-evaporative
mass loss using steady-state one-dimensional wind models.
They found that only in rare cases is the planetʼs orbit affected
by wind torques.
7
To get a sense for the magnitude of the spin synchronization timescale we
use Equation (10) in Matsumura et al. (2010) and Ωpl(t) = Ωo(t − Δt), where
Δt is the time interval between two consecutive time steps during an orbital
evolution calculation.
4
The Astrophysical Journal, 813:101 (13pp), 2015 November 10
Valsecchi et al.
planets. We further discuss tidal locking for the planet in
Section 5.
Even though our calculations account for tides in both
components, our results show that only stellar tides can affect
the orbital separation significantly. In particular, for a slowly
spinning stellar host, tides transfer angular momentum from the
orbit to the stellar spin, causing orbital decay.
For comparison, the evolution of the orbital separation due to
RLO is given by
M˙ pl,RLO
⎛ a˙ ⎞
⎜ ⎟
(dg - 1) > 0.
~ 2
⎝ a ⎠RLO
Mpl
This model assumes that a fraction δ of M˙ pl,RLO settles into a
ring whose radius ar is a constant multiple γ2 of the orbital
separation a. This mass is then lost from the system, taking
with it the specific angular momentum of the ring. The
remaining fraction of the mass (1–δ) is assumed to be accreted
onto the host star. Below we describe our choices for γ and δ,
based on the expected stability of MT.
3.3. Magnetic Braking
For the loss of stellar spin angular momentum via MB we
follow Skumanich (1972) and adopt
( W˙ *)MB = -aMB W*3 ,
(10)
(8 )
3.4.3. Considerations on the Stability of MT
−14
where αMB = 1.5 × 10
year (e.g., Dobbs-Dixon et al. 2004;
Barker & Ogilvie 2009; Matsumura et al. 2010; Valsecchi &
Rasio 2014a, 2014b). This law is well established for the stellar
equatorial rotation rates of interest here (1–30 km s−1).
The calculations presented in Section 4 show that conservative MT is always stable. Instead, some care must be taken
when considering non-conservative MT.
For the systems under consideration, the stability of the MT
phase depends mainly on the fraction of planetary mass leaving
the system, its specific angular momentum, and the response of
the planet to mass loss. For a mass-losing planet the latter
cannot be determined a priori and it is computed with MESA as
the orbital evolution proceeds. We also have no prior
knowledge of the values of γ and δ (the parameters regulating
the amount of mass leaving the system and how much specific
angular momentum is carried away) that correspond to
dynamical stability of MT. However, some guidance on the
region of stability can be gained as follows. For systems with
extreme mass ratios (q = 1, such as those considered here) and
J˙orb Jorb given by Equations (6) and (9), the planetary mass
during RLO changes according to (e.g., Rappaport et al. 1982
for a derivation)
3.4. Roche-lobe Overflow
RLO is modeled by implicitly computing the MT rate that is
required for the planetary radius to remain below its Roche lobe
radius, for which we use the approximation by Eggleton
(1983). This procedure is described in Section 2.3.2 of Paxton
et al. (2015). Here we consider both conservative and nonconservative MT. When MT is conservative, all mass leaving
the planet via RLO is accreted onto the star and
M˙ * = -M˙ pl,RLO. Instead, during non-conservative MT evolution, some fraction δ of M˙ pl,RLO is lost from the system and
M˙ * = -(1 - d ) M˙ pl,RLO. Note that in none of the examples
presented here does the star fill its Roche-lobe during the
orbital evolution calculation.
M˙ pl,RLO
Mpl
3.4.1. Stable Conservative MT
During conservative MT, the evolution of Jorb is generally
computed under the assumption that MT proceeds through an
accretion disk. In the standard picture, the matter flowing
through the inner Langrangian point appears to be pushed into
orbit about the host star by Coriolis forces (as viewed in the
corotating frame). Subsequently, viscous stresses spread this
material into a disk around the accretor (Frank et al. 1985). This
disk transports mass toward the accretor and angular
momentum away from it. The latter is eventually returned to
the orbit via torques operating between the donor and the outer
edge of the disk (Lin & Papaloizou 1979) and a small residual
angular momentum is transferred to the spin of the star. Our
calculations with MESA include spin-up through accretion.
However, as the material accreted by the star is only a tiny
fraction of its total mass, spin-up due to accretion is negligible.

During non-conservative MT, changes in Jorb are computed
as in Soberman et al. (1997)
2
M˙ pl,RLO
Mpl
 dg
M˙ pl,RLO
Mpl
.
x
5
- dg +
2
6
. (11)
Here (R˙pl Rpl )therm is the fractional rate of change of the planet
radius due to its thermal evolution and (J˙orb Jorb )tides is the
fractional rate of change in orbital angular momentum due to
tides. Finally, x = (d ln Rpl d ln Mpl )ad is the planetʼs adiabatic logarithmic derivative of radius with respect to mass, not
to be confused with the mass–radius relations in Figure 1, valid
for thermal equilibrium models. A necessary condition for
stable MT is that the denominator of Equation (11) be positive.
Our goal here is to improve on the stable MT calculations
presented in Valsecchi et al. (2014) by investigating the
importance of irradiation effects and self-consistent models for
the planet. To explore MT stability in the present work, we set
δ = 1 and consider γ values for which the MT is expected to be
dynamically stable. There is no loss of generality in this
prescription since γ and δ appear only as a product in
Equation (11). We discuss possible realistic MT configurations
in Section 5, while reserving a more detailed analysis of MT
stability in hot-Jupiter systems to a future study. Such analysis
should account for a broad range of γ and δ values, as well as
3.4.2. Stable Non-conservative MT
⎛ J˙orb ⎞
= dg (1 + q)1
⎜
⎟
⎝ Jorb ⎠RLO
⎛ J˙ ⎞
M˙ pl,PE ⎛ 1
x⎞
1 ⎛ R˙ pl ⎞
⎜
⎟
- ⎜ orb ⎟
+ ⎜
- ⎟
⎝ Jorb ⎠tides
2 ⎝ R pl ⎠therm
Mpl ⎝ 6
2⎠
(9 )
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Valsecchi et al.
initial orbital configurations and properties of the components
(e.g., M*, Mpl, and metallicity; see Section 6).
With δ = 1 and γ ¹ 0 we are assuming that all RLO material
leaves the system carrying away a specific angular momentum
equal to g GM* a . The condition for stability, i.e., the
requirement of a positive denominator in Equation (11), then
reduces to
g<
5
x
+ .
6
2
(12)
Equation (12) shows that an increase in the planetary radius
with adiabatic mass loss (i.e., ξ < 0) has a destabilizing effect,
as expected. We test different values of γ with MESA and find
that, depending on the planetary core mass, Mc, the computation becomes numerically difficult for γ values higher than
about 0.6–0.8. This suggests that the MT may indeed become
dynamically unstable and that, according to the criterion in
Equation (12), ξ is inferred to be close to zero. To avoid the
instability region, while still considering somewhat significant
angular momentum loss from the system, we provide examples
for non-conservative MT with γ = 0.5 and γ = 0.6 for the 1 M⊕
and 5 M⊕ core cases, respectively, and γ = 0.7 for the higher
core masses.
Figure 3. Detailed evolution of some of the components and orbital parameters
(top) and of the relevant timescales (bottom) for a Jupiter with Mc = 5 M⊕. For
this evolution we adopt the parameters δ = 1 and γ = 0.6. The subscripts
“RLO,” “PE,” and “Tides” refer to the timescales associated with mass loss due
to Roche-lobe overflow and photo-evaporation, and to tidal decay, respectively.
For the latter we note that only stellar tides play a significant role, as the
planetary tides timescale is always longer than a Hubble time. The peaks in the
tidal timescales occur when the contributions from stellar and planetary tides
cancel out. The dotted blue curve in the upper panel indicates the Roche-lobe
radius, while the dotted curve in the bottom panel is the stellar nuclear
timescale. The RLO phase ends after about 1.6 Gyr, when the system is about
3.6 Gyr old. For clarity, the timescale for the evolution of the planetary radius
(bottom panel, green line) is computed taking the median of 20 consecutive
values. The calculation ends when Menv/Mpl  7% because of convergence
problems.
4. RESULTS
We consider a typical hot-Jupiter system comprising a 1 Me
star and a 1 MJ planet at solar metallicity. For the planet we
took Mc = 1 M⊕, 5 M⊕, 10 M⊕, 15 M⊕, and 30 M⊕. The initial
period was set to Porb = 0.7 day in all cases. This corresponds
to the orbital period at which a hot Jupiter with a 1.5 RJ radius
would be at its Roche limit and it is chosen arbitrarily to have
all systems starting with the same initial period “close enough”
to the Roche limit.
For the initial systems’ age we chose 2 Gyr (t ; 20% tMS), as
it is at the low end of the ages of the currently known hot
Jupiters closest to their Roche-limit separation (see Table 2 in
Valsecchi & Rasio 2014a and references therein; see also
Section 5). Accordingly, we create 2 Gyr old stellar and
planetary models with MESAʼs “single-star” module. Each
planet is irradiated according to the host star properties at 2 Gyr
and the 0.7 day period (F* = 5.4 × 109 erg cm−2 s −1).
The models are then used in MESAʼs “binary” module to
compute the orbital evolution. During this step both stellar and
planetary evolution, as well as irradiation and PE are computed
self-consistently (Section 2). For the stellar spin, we choose an
initial value of Ω* ∼ 0.1 Ωo, where Ωo is the orbital frequency.
This is consistent with the observed slow stellar rotation rates
for the tightest hot-Jupiter systems known (Ω* ; 0.1–0.2 Ωo;
see Table 1 in Valsecchi & Rasio 2014a). Finally, as described
in Section 3.4, we consider both conservative and nonconservative MT.
The results are presented as follows. In Section 4.1 we
describe in detail the evolution of the 5 M⊕ and 30 M⊕
planetary core cases. We take these as extreme examples
among those considered here. In fact, as described below
(Section 4.2), the behavior of the 1 M⊕ core model needs a
more in-depth investigation. For Mc = 5 M⊕ and Mc = 30 M⊕,
we mainly focus on the non-conservative MT evolution and
briefly describe the conservative case at the end of each section.
For these same core masses, we also investigate the effect of
varying the column density for irradiation in Section 4.1.3, and
the PE recipe in Section 4.1.4. In Section 4.2 we present an
overview of how the orbital period evolves as the planet loses
mass, as well as how the planetary radius evolves with mass,
for the full range of core masses. A summary of the results for
all core masses and physical assumptions is presented in
Table 2. In all examples considered, the evolution is quite rapid
after the planet is left with only a few percent of the envelope
mass. Thus, we discuss the final stages of the planetary
evolution qualitatively, guided by the relevant timescales
entering the problem.
4.1. Detailed Examples: Different Core Masses
4.1.1. The 5 M⊕ Core Model
Figure 3 shows the evolution of a Jupiter with a 5 M⊕ core
undergoing non-conservative MT. The top panel displays the
evolution of various system and planetary properties, while
the bottom panel investigates a number of different timescales
of the system. For convenience in studying the plot, we
arbitrarily subtract 2 Gyr from the time axis (this is just the
time over which we evolved the star and the planet before
inserting them into the binary evolution version of the MESA
code; see discussion above). Hereafter, when describing
various features in Figure 3 (as well as in Figure 4) we refer
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Valsecchi et al.
(2) of Howell et al. 2001):
-1 2
Porb  0.4 ( Mpl MJ )
( Rpl RJ )3 2
days.
(13)
Thus, if the MT remains stable, then the orbital period will
grow or shrink in accordance with how the planetary radius
changes with mass loss. In this example with a 5 M⊕ core, we
can see from Figure 3 that the planetʼs radius does not vary
very much during most of the RLO phase, and in fact, slightly
increases to 1.6 RJ. Therefore, Porb will simply grow mostly as
µMpl-1 2 . By the end of the RLO phase the planetʼs mass has
shrunk to 8.5 M⊕, and, according to Equation (13) Porb should
be about 4 days, which it is. The RLO phase ends when the
combination of tidal decay of the orbit and planetary expansion
is no longer sufficient to keep the planet filling its Roche lobe,
and the planet, which is now quite low in mass, starts to shrink
well inside its Roche lobe.
The post-RLO phase is driven mainly by PE. In fact, the tidal
decay timescale becomes longer than a Hubble time for the
majority of this phase due to the longer orbital period and the
greatly reduced planetary mass. PE removes mass from the
planet at a rate of about 10−16–10−15 Me yr−1 (1010–
1011 g s−1) and it does not affect the orbital separation
significantly (see Equation (7)). For the 5.4 Gyr duration of
the post-RLO phase the planet remains in a 3.9 day orbit and its
mass decreases from about 8.5 M⊕ to about 5.4 M⊕ (when
Menv/Mpl ; 7%), when the calculation ends because of
convergence problems.
The bottom panel of Figure 3 displays various timescales for
this evolution, including the mass loss timescales via the photoevaporative wind, ∣ Mpl M˙ pl ∣PE , and due to RLO, ∣ Mpl M˙ pl ∣RLO ,
as well as the timescale for tidal decay of the orbit,
∣ Jorb J˙orb ∣tides , and for the thermal expansion/contraction of
the planetary radius, ∣ Rpl R˙pl ∣. What we see is that, after RLO
commences, the black curve, showing the MT timescale (due to
RLO), lies a factor of about 8 lower than the tidal decay
timescale, and much lower than the mass loss timescale
associated with PE winds. We learn from Equation (11) that the
mass loss rate due to RLO is therefore essentially proportional
to the rate of decay of the orbit due to tides, and inversely
proportional to the denominator which is ξ/2 − γδ + 5/6 = ξ/
2 + 0.23. Putting these together implies that the denominator
must equal approximately 1/8. From this we can infer that the
effective adiabatic index of the planet during most of the RLO
phase is ξ ; −0.2, in other words the planet reacts to adiabatic
mass loss by slightly expanding. Later in the RLO phase, the
tidal decay timescale greatly increases due to the increasing
orbital separation, but the thermal expansion timescale of the
planet decreases dramatically to pick up the slack of the
declining tidal effects. We also see that the mass loss rate in a
PE driven wind dramatically increases (i.e., the timescale
decreases) due to the increasing radius and the decreasing mass
of the planet (see Equation (4)).
Even though our evolution calculation ends when the
envelope mass fraction drops to about 7%, we argue that
eventually, the host star will approach the end of its mainsequence lifetime and it will begin expanding. The increase in
R* will cause a concomitant increase in both the tidal decay
rate and in the amount of irradiation received by the planet. The
latter occurs because R* evolves faster than T* in Equation (1).
Such an increase in irradiation will yield an increase in Rpl and,
as a consequence, even stronger PE. What happens to this
Figure 4. Same as Figure 3, but for a Jupiter with Mc = 30 M⊕ and γ = 0.7.
For clarity, ∣ Mpl M˙pl ∣RLO is computed by taking the median of 10 consecutive
values.
to the time marked on the axis, i.e., after subtracting off a
2 Gyr reference time.
For the first ∼70 Myr the planet underfills its Roche lobe
while the orbit shrinks due to the tides transferring angular
momentum from the orbit to the stellar spin, in a vain attempt
to try to spin-up the star. Eventually, when Porb shrinks to
;0.5 day (a ; 0.01 AU) the planet fills its Roche-lobe. This
occurrence can be seen when the dotted curve in Figure 3
(indicating the Roche lobe radius) merges with the planetary
radius curve (solid blue).
The system remains in Roche-lobe contact for the ensuing
1.6 Gyr, while the planet loses more than 95% of its mass.
Concomitantly, the orbital period grows from about 0.5 to 3.9
days. During this same interval, the effective temperature of the
planet decreases from about 2500 K to about 1300 K, due
largely to the increasing distance between the host star and the
planet.
During the RLO phase there is a competition between the
tidal effects, which tend to drive orbital decay, and MT, which
generally drives orbital expansion (see Equation (10)). Both
effects operate simultaneously, and the one that dominates
depends on the response of the planetʼs radius to mass loss. For
Roche-lobe overflowing objects with masses lower than the
accreting star, there is a universal relation between the orbital
period, and mass and radius of the donor (see, e.g., Equation
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Valsecchi et al.
system after the envelope of the planet has been removed,
leaving only its rocky core, depends mainly on the tidal
timescale compared with the stellar nuclear evolution timescale. The former determines the rate of tidal decay (leading to
a second planetary RLO), while the latter determines the rate
of stellar expansion (leading to stellar RLO). Given the (very
steep) dependence of the tidal decay timescale as (a R* )8 (e.g.,
Valsecchi & Rasio 2014b), we argue that tides will likely drive
the planet down to its Roche limit. Here, it will be stripped of
the remaining envelope and, assuming a constant density
for the core, it will be consumed at approximately constant
Porb on the rapid tidal decay timescale. This follows from
Paczyńskiʼs (1971) approximation for the Roche limit separation aR = (Rpl 0.462)(M* Mpl )1 3 µ (M* rpl )1 3 , where ρpl
denotes the mean density of the planet.
The above discussion was for the case of non-conservative
MT (with γ = 0.6). In the case of conservative MT, we find that
the orbital evolution proceeds similarly to the examples
presented above, but the duration of the various phases is
different. In fact, when MT is conservative, the tidally driven
orbital decay is counteracted by a more substantial RLO-driven
orbital expansion [δ = 0 in Equation (10)], which results in a
slower overall evolution. This is summarized in Table 2. In
particular, the RLO phase lasts for about 4.5 Gyr, leaving an
8.3 M⊕ planet in a 3.4 day orbit. The remainder of the evolution
proceeds similarly to the non-conservative case. However, the
evolution when little envelope mass is left is shorter. This is
due to the irradiation-driven decrease in planetary mass loss
timescale (∣ Mpl M˙ pl,PE ∣) due to the star approaching the end of
its main sequence.
We can gain some further insight into the evolution of this
system by considering the timescales displayed in the bottom
panel of Figure 4. For most of the evolution, the tidal decay
timescale (∣ Jorb J˙orb ∣tides) is about 6 times longer than the mass
loss timescale (∣ Mpl M˙ pl ∣RLO ). Furthermore, at least for the
earlier portion of the RLO phase, both the contraction timescale
of the planet (∣ Rpl R˙pl ∣) and the timescale of PE (∣ Mpl M˙ pl ∣PE )
are longer yet than the tidal decay timescale. From this, we can
infer that the denominator in Equation (11) must be ;1/6. In
turn, we can conclude that ξad ; 0.07 (i.e., very close to zero).
During this earlier portion of the RLO evolution, the orbital
period grows and the mass of the planet declines, a
combination that leads to an ever-increasing tidal decay
timescale (i.e., weakening of the tidal evolution of the orbit).
During the later portion of the RLO phase, both ∣ Rpl R˙pl ∣ and
∣ Mpl M˙ pl ∣PE become shorter than the tidal driving timescale.
This means that there is close competition in the numerator of
Equation (11) for maintaining Roche-lobe contact among all
three terms: (i) tidal decay; (ii) the photo-evaporative mass loss
term (tending to shrink the Roche-lobe radius of the planet and
maintain RLO), and (iii) the shrinkage of the planet as it loses
mass, which tends to push the planet back within its Rochelobe. Apparently, the combination is sufficient to maintain
RLO until we terminate the evolution.
Our calculation ends when the planetary envelope has been
completely removed. However, the core itself may well
eventually undergo RLO and, if the core has a constant
density, it will be consumed at constant orbital period on the
rapid tidal decay timescale (100 Myr, at the end of the
evolution in Figure 4).
The conservative MT case for the same model with a 30 M⊕
core differs only insofar as the duration of the RLO phase is
concerned (see Table 2). This is the same as what we found for
the conservative versus non-conservative cases with a 5 M⊕
core. In fact, during conservative MT, the RLO phase until the
envelope is removed lasts about 3.8 Gyr. At the end of the
calculation Porb = 0.3 day.
4.1.2. The 30 M⊕ Core Model
Figure 4 shows the evolution of a Jupiter with a 30 M⊕ core
undergoing non-conservative MT (with γ = 0.7). As in
Figure 3 (for the case of a 5 M⊕ core), the top panel presents the
evolution of various system and planetary properties, while the
bottom panel displays a number of different timescales of the
system. For the first ∼70 Myr, tides cause the orbit to shrink
and the planet fills its Roche lobe when Porb ; 0.5 day (the
same as for the 5 M⊕ core case).
At the onset of RLO, the orbit expands as mass is removed
from the planet at a rate of ∼10−13–10−12 Me yr−1 (1013–
1014 g s−1). However, after about a Gyr, the orbit begins
shrinking once the period has grown to only ∼0.8 day. Overall,
we follow the RLO phase for about 2.1 Gyr, to the point where
the planetary envelope has been completely removed. At this
time, the orbital period has shrunk to 0.3 day, and the effective
temperature of the planet has increased to nearly 3000 K.
The difference in evolutionary history between this planet
with a 30 M⊕ core and the one with a 5 M⊕ core (described in
Section 4.1.1) results largely from the fact that planets with
more massive cores exhibit an earlier (i.e., at higher Mpl;
Figure 6) decrease in planetary radius with continuing mass
loss. During the RLO phase with a 30 M⊕ core, the mass of the
planet decays essentially as a power law in time, while the
radius does not decrease significantly until Mpl drops below
∼60 M⊕. From Equation (13) we can deduce that this
combination of mass and radius changes will lead to a steady
increase in the orbital period. However, once the radius starts to
decline substantially, the Rpl3 2 dependence in Equation (13)
dominates the orbital period evolution, and Porb starts to decay.
4.1.3. Varying the Column Density for Irradiation
For highly irradiated planets with Mpl < MJ, the quasiequilibrium mass–radius relations of Fortney et al. (2007) in
Figure 1 are bracketed by those computed with MESA for
irradiation
absorption
column
densities
of
Σpl = (1–100) g cm−2. We tested the effect on planetary RLO
evolution by increasing Σpl by two orders of magnitude in the
5 M⊕ and 30 M⊕ core mass models. We find that the overall
evolution does not change significantly (see Table 2). Qualitatively, for the 5 M⊕ core case, at the beginning of the
calculation the radius at higher Σpl is a few percent larger.
As a result, RLO starts at a longer orbital period and, for the
same evolutionary time, it continues at longer Porb. When the
planet retreats back within its Roche-lobe, the longer orbital
period yields a less severe mass loss via PE. Consequently, the
planet retains a higher envelope mass fraction (f in Table 2) for
a longer time. A similar behavior occurs for the 30 M⊕ core
model, but the percent difference in radius (orbital period)
increases from ∼5% to ∼15% (from ∼5% to ∼20%) between
the beginning and the end of the calculation. Again, mass loss
proceeds on a longer timescale and, for the same evolutionary
time, the models with higher Σpl retain a larger fraction of the
envelope for longer.
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Valsecchi et al.
4.1.4. Varying PE
Formally, the prescription of Murray-Clay et al. (2009) uses
FXUV ∼ 104 erg cm−2 s−1 as a threshold between the energylimited and radiation/recombination-limited regimes. Below
we investigate whether our results for the 5 M⊕ and 30 M⊕ core
cases change significantly by using Equation (3) when
FXUV > 104 erg cm−2 s−1 and Equation (4) otherwise. The
results are summarized in Table 2 (denoted with “e-Lim +
rr-Lim” in the column named “PE”).
The planetary mass loss rate in the radiation/recombinationlimited regime is slower. This naturally leads to a longer
duration of the RLO phase for all examples considered. For the
5 M⊕ core undergoing conservative MT, Rpl is a few percent
larger. As a result, the orbit evolves at longer period. However,
because of the longer RLO phase, by the time the planet is left
with about 10% of its envelope mass, the star is approaching
the end of its main sequence. Consequently, the increase in Rpl
driven by the increase in stellar irradiation leads to faster mass
loss via PE toward the end of the calculation. For the 5 M⊕ core
undergoing non-conservative MT, the evolution follows this
same line of logic, but the planetary radius is a few percent
smaller for most of the evolution. As a result, the orbit evolves
at shorter period. For the 30 M⊕ core case, the response of Rpl
to mass loss for the different PE prescriptions differs by only
1%–2%. Therefore, only the duration of the RLO phase
changes significantly, while the orbital period at the various
stages of Table 2 is not significantly affected.
In summary, none of our basic results would change
significantly if we were to utilize both the energy-limited and
radiation/recombination-limited prescriptions for PE over the
entire evolution. However, certainly a number of the details of
the evolutionary models would look somewhat different.
Figure 5. Planetary mass as a function of the orbital period (top) and mass–
radius relation (bottom) for conservative MT (with δ = 0). Because some of the
evolutionary calculations become numerically challenging when the mass
fraction in the envelope drops below a few percent, we show the evolution up
to when Menv/Mpl drops just below 5%. The open orange circles are confirmed
exoplanetary systems (NASA Exoplanet Archive, 2015 January 13) with
observationally inferred Mpl and Porb, and hosting one planet only, for
simplicity. The system GJ 436b (Butler et al. 2004) would be located at
Porb ; 2.6 d and Mpl ; 22 M⊕. The colored lines are our evolutionary models
for different core masses. The colored arrows along each evolutionary track in
the top panel mark 1 Gyr intervals and denote time evolution. For the
remaining symbols:“×s” mark the end of RLO, “,s” mark times when the
mass in the envelope drops below f = 30%, and 5% of the total mass. For the
5 M⊕ core model, the various symbols in the top panel are all superimposed at
the end of the evolution, where the planet spends a few Gyrs.
4.2. A Range of Evolutionary Models
A range of evolutionary models covering different core
masses for both conservative and non-conservative MT are
presented here. The results are summarized in Figures 5 and 6,
respectively. The top panels show the evolution of the orbital
period with the planetʼs mass. For comparison, the positions of
the observed planets are the orange open circles. The bottom
panels show the corresponding evolutionary tracks for
planetary mass and radius. The evolution tracks for different
core masses are denoted with different colors and line-styles.
For all cases considered, the MT proceeds on a timescale
longer than the thermal timescale of the planet. Thus, the planet
remains in near thermal equilibrium throughout the MT. Here
we note that our 1 M⊕ core model shows a severe increase in
Rpl that needs further investigation. In fact, Rpl increases up to
about 10 RJ when Mpl ; 1 M⊕. However, we note also that the
coreless models in Figure 8 of Fortney et al. (2007) have radii
ranging up to Rpl ; 2.3 RJ when Mpl ∼ 30 M⊕ (the lowest
mass considered by Fortney et al. (2007) for a coreless
Jupiter), depending on the age of the planet and the level of
irradiation. This is consistent with our radii for masses
down to ∼10 M⊕. We omit the 1 M⊕ core model from the
subsequent discussion.
The evolution in time in the Mpl–Porb plane is downward
(i.e., shrinking mass), initially toward longer periods, but then
decaying toward shorter Porb. The tracks in the (Rpl–Mpl)
diagram proceed from the right to the left. The shape of the
evolutionary tracks in such diagrams depends on the
competing effects of stellar tides (tending to shrink the orbit)
and RLO (tending to expand the orbit). As described in
Section 4.1.1, the Rpl–Mpl tracks in the bottom panels of
Figures 5 and 6 partly determine which mechanism ends up
dominating in terms of net orbital contraction or expansion
(see Equation (13)).
At the onset of RLO, the nearly constant or increasing Rpl
with decreasing Mpl causes the RLO term to dominate over the
tidal contribution. As a result, the orbit expands. This behavior
persists until the mass–radius relation begins steepening,
becoming more positive. At this point, the tidal term becomes
more significant than the RLO term and the orbit begins to
shrink. In some cases (Mc = 5 M⊕, 10 M⊕, 15 M⊕), the
combination of the planet shrinking in response to mass loss
and the weakening tidal forces with decreasing Mpl causes the
RLO phase to terminate and the planet to detach (“×”
symbols). In a few cases (Mc = 5 M⊕, 10 M⊕ during
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5. DISCUSSION
Our calculations seem very promising for explaining some of
the super-Earth and sub-Neptune-type planets whose bulk
density suggests that they consist of a core (rocky or icy)
surrounded by a H/He envelope comprising up to tens of a
percent of the total mass (Lopez & Fortney 2014). While this
agreement with our MT models is very encouraging, our
calculation neglects some important effects, which we discuss
below. Specifically, in Section 5.1 we discuss possible MT
scenarios, while in Section 5.2 we discuss our assumptions on
planetary tides. A discussion of observational signatures is in
Section 5.3.
5.1. MT Scenarios and Stability
In this work we neglected the effects of magnetic fields and
stellar winds on the RLO material. These mechanisms can
affect the flow of MT (e.g., Cohen & Glocer 2012; Owen &
Adams 2014), potentially playing a crucial role in determining
whether any mass is transferred or an accretion disk ever forms.
On the opposite side of conservative MT is the case where
mass is directly blown away from the planet or the inner
Lagrangian point (L1). For a 1 MJ planet and a 1 Me star, L1 is
located at ;0.93 a (Equation (10) of Lai et al. 2010). In the
formalism of Section 3, these scenarios can be reproduced by
setting δ = 1 and γ = 1 (mass loss from the planet) or γ = 0.97
(mass loss from L1). In this configuration, Equation (12)
requires ξ to be larger than (0.27–0.33) in order for the MT to
be dynamically stable. The values of ξ inferred in Sections 4.1.1
and 4.1.2 for our extreme core masses (ξ = −0.2 and 0.07 for
the 5 M⊕ and 30 M⊕, respectively) suggest that the MT will
likely be dynamically unstable if γ  0.97. Furthermore, we
have performed test runs with MESA fixing δ = 1 while
varying γ. We find that the MT may become dynamically
unstable for γ … 0.8. In fact, the computation becomes
numerically difficult, with the integrator time step dropping
to less than one month. This limit on γ for stability suggests
that the effective value of ξ is close to zero, or negative (as
directly computed within MESA).
To gain better insight into more physically meaningful
values of the parameter γ, consider the case where matter flows
in a narrow stream from the L1 point, and forms a ring around
the star, which is then blown away. This case has been
discussed in the context of MT stability in stellar binary
systems (e.g., Hut & Paczynski 1984; Verbunt & Rappaport 1988). If, in fact, the lifetime of the gas in the ring, before
being blown away, is shorter than the viscous timescale for the
ring, then the angular momentum may not be returned to the
orbit. In this case, the angular momentum leaving the system
would be determined by how much angular momentum a
particle has before it is blown away (i.e., by the mean ring
radius rd). We combine calculations of rd by Lubow & Shu
(1975) and Hut & Paczynski (1984), covering Mpl/M* down to
10−3, with our own calculations, extending these down to Mpl/
M* = 10−6. We find rd to range between ;0.7–0.9 for a 1 MJ −
10 M⊕ planet. This range implies γ values between ;0.84–0.95
and a value of ξ for stability between ;0−0.23 (Equation (12)).
This scenario appears borderline between stable and unstable
MT. Clearly any scenario where γ > 1 would be dynamically
unstable.
If the MT were indeed dynamically unstable, one could
envision the system undergoing a common-envelope-like
Figure 6. Same as Figure 5, but for non-conservative MT (with δ = 1). We use
γ = 0.5, 0.6, and 0.7 for Mc = 1 M⊕, 5 M⊕, and …10 M⊕, respectively. Note,
for the 5 M⊕ core model, the evolution terminates because of convergence
problems when the envelope contains ;6% of the total mass. For the 5 M⊕
core model described in detail in the text, the various symbols in the top panel
are all superimposed at the end of the evolution, where the planet spends
few Gyrs.
conservative MT), as the star approaches the end of its main
sequence, the increasing amount of irradiation received by the
planet causes Rpl to increase again (e.g., Fortney & Nettelmann 2010). Note that none of the observed exoplanets in
Figures 5 and 6 are currently in RLO. Therefore, observations
should be compared with the portion of the evolutionary tracks
where the planet is detached. There are some 6–7 observed
systems shown in these figures that are in the vicinity of our
evolution tracks, after RLO has stopped, and our models may
be directly applicable to them. Note also that Ehrenreich et al.
(2015) recently reported the discovery of a large exospheric
cloud surrounding the Neptune-mass exoplanet GJ 436 (Butler
et al. 2004), composed mainly of hydrogen atoms. The average
observed mass-loss rate implies an efficiency for converting
X-ray and extreme UV energy into mass loss of about 1%. In
Figures 5 and 6 this planet would be located at Porb ; 2.6 day
and Mpl ; 22M⊕.
Finally, we note that the shape of the evolutionary tracks
does not change significantly in the Mpl–Porb diagram between
conservative and non-conservative MT. However, as summarized in Table 2 and explained in Section 4.1, whether mass is
lost from the system or not does affect the duration of the
various phases mentioned above.
10
The Astrophysical Journal, 813:101 (13pp), 2015 November 10
Valsecchi et al.
evolution (“CE”), that is somewhat distinct from the standard
binary stellar evolution picture (Webbink 1984). We envision
that the envelope of the planet would quickly flow though the
inner Lagrange point and form a disk-like structure around the
host star. The core of the planet would then find itself orbiting
within this ring or disk of envelope material. Depending on the
detailed core–disk interactions, the core may spiral-in toward
the host star and eject the disk material. This is distinct from the
usual CE scenario in that the envelope of the planet becomes
bound to the host star, rather than the planet, and it is the planet
which ejects its own remnant envelope via tidal and viscous
interactions. Insights into the outcome of such a phase can then
be gained from the energy equation
⎡ GM Mc
GM* Mc ⎤
GM* Menv
*
a CE ⎢
.
⎥
⎣ 2a f
2a i ⎦
2a i
These include potentially significant heating, inflation, and
resultant stronger mass loss. To gain some intuition into their
significance, we compute the power deposited between two
consecutive integration steps and compare it with the planet
luminosity, Lpl, at each integration step. For the former, we use
1
L tide = Ipl ∣ Wo2 (t + Dt ) - Wo2 (t )∣ Dt , where Ipl is the
2
moment of inertia. We find that Ltide/Lpl decreases from 10−4
to 10−11 throughout the calculation. This suggests that the
power deposited in the planet via tides may not affect its
structure dramatically.
5.3. Observational Signatures
As discussed in Valsecchi et al. (2014), this evolutionary
scenario has several observational consequences. First, the
properties of stars hosting hot Jupiters and those hosting SuperEarth and mini-Neptune-type planets should be similar. For
example, if all RLO material is lost from the system, the host
stars should have similar mass and composition, but superEarth and mini-Neptune host stars should be somewhat more
evolved. Furthermore, depending on the core mass and the
details of the MT process (whether any mass is lost from the
system and the location where angular momentum is removed),
a system may spend enough time in RLO that it might be
possible to observe planets in such a phase. If MT really
proceeds through an accretion disk, this may produce
observational signatures (e.g., line absorption of stellar
radiation and time-dependent obscuration of the starlight; Lai
et al. 2010). Finally, the results in Figures 5 and 6, and in
Table 2 suggest that, if this model is a viable formation channel
for super-Earth and mini-Neptune-type planets, there should be
a correlation between Mpl and Porb. Specifically, the more
massive planets should be found at shorter orbital periods. This
is a major result when compared to the simple model of
Valsecchi et al. (2014). In fact, without a self-consistent
calculation of planetary evolution and irradiation effects for a
spectrum of core masses, we had found no trend between final
Mpl–Porb pairs. The different selection of core masses adopted
in Valsecchi et al. (2014) does not allow for a one-to-one
comparison of those results with the findings of this paper.
However these different results can be understood simply via
the way in which the orbital separation evolves in response to
planetary mass loss, and via the incomplete description of the
pre-determined mass–radius relations utilized in Valsecchi
et al. (2014). The latter fixed Mpl–Rpl relations are shown in
Figure 7, compared with those computed with MESA during
the course of the evolutions.
This inverse correlation between orbital period and planetary
mass that we have found also suggests that dynamically stable
MT phases like those presented here do not seem to represent a
viable channel for the formation of the so-called “ultra-shortperiod planets” (e.g., “USPs”; Sanchis-Ojeda et al. 2014 and
reference therein). These planets have typical radii smaller than
2 R⊕ (corresponding to a mass of about 5 M⊕; Weiss &
Marcy 2014) and orbital periods shorter than one day.
(14)
Here, αCE represents the efficiency with which the planet coreʼs
orbital energy can be used to unbind the envelope material,
which is now in a ring around the host star. The parameters ai
and af denote the orbital separation at the onset and at the end
of this “CE” phase, respectively. Equation (14) can be directly
solved for the ratio of af/ai, and we find
af
ai

a CE Mc
.
a CE Mc + Menv
(15)
For plausible values of αCE near unity, this expression yields
af/ai ; Mc/Mpl. This, in turn, implies that the orbital separation
would decay by a factor of more than an order of magnitude if
the MT is unstable at the onset of RLO.
Thus, all cores considered in this work would reach their
own Roche limit in the event of unstable RLO.8 A new RLO
phase would then begin driven, this time, by the interplay of
viscous drag on the core due to the remaining envelope mass
orbiting the host, and the back-reaction from continuing (now
stable) MT through RLO. The former would tend to shrink the
orbit, while the latter would cause the orbit to expand.
The actual outcome of an unstable MT phase from a hot
Jupiter to its stellar host is still an unexplored question that we
are currently addressing via smoothed-particle hydrodynamics
simulations. Furthermore, magnetohydrodynamics (MHD)
simulations are needed to determine the role played by
magnetic fields. Here we note that Matsakos et al. (2015)
recently investigated magnetized star–planet interactions via
3D MHD numerical simulations including stellar wind and
photo-evaporative mass loss. They found that, depending on
the planetʼs magnetic field and outflow rate, as well as the
stellar gravitational field, the star can accrete part of the mass
lost by the planet; Pillitteri et al. (2015) proved this to be a
plausible scenario via far-ultraviolet observations of HD
189733.
5.2. Planetary Tides
In this work we have taken planetary tides to be efficient in
keeping the spin of the planet tidally locked. This assumption is
justified by the magnitude of the tidal synchronization timescales discussed in Section 3. However, we neglected the
resulting effects of tidal dissipation on the planetary structure.
6. CONCLUSIONS
In this paper we have presented the first evolutionary
calculations of irradiated hot-Jupiters undergoing tidally driven
RLO. We found that, depending on the size of the planetary
core and the details of the MT, the RLO phase and, in turn, the
detached phase (after RLO had ceased) can last from a few to
8
If R* is too large, the planet-core will plunge into its atmosphere before
filling its Roche lobe; the critical period for tidal breakup of a rocky body can
be 5 hr (Rappaport et al. 2013).
11
The Astrophysical Journal, 813:101 (13pp), 2015 November 10
Valsecchi et al.
determine whether hot Jupiters can undergo the kind of stable
RLO MT described in this work.
We thank the anonymous referee for many very helpful
comments. F.V. and F.A.R. are supported by NASA Grant
NNX12AI86G. F.V. is also supported by a CIERA fellowship.
L.A.R. gratefully acknowledges support provided by NASA
through Hubble Fellowship grant #HF-51313 awarded by the
Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc.,
for NASA, under contract NAS 5-26555. We thank Dorian
Abbot, Arieh Konigl, Titos Matsakos, Ruth Murray-Clay,
James Owen, and Dave Stevenson for useful discussions. This
work used computing resources at CIERA funded by NSF
PHY-1126812. This research has made use of the NASA
Exoplanet Archive, which is operated by the California
Institute of Technology, under contract with the National
Aeronautics and Space Administration under the Exoplanet
Exploration Program.
Figure 7. Mass–radius relations. In black are the planetary models computed in
this work, while in red are those used in Valsecchi et al. (2014).
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